2
$\begingroup$

Let $X$ be a normal projective variety over the field of complex numbers. Let $Y$ be a subvariety of $X$, and let $I_Y$ be the ideal sheaf of $Y$. From what I know, I can define the blow-up of $X$ along $Y$ as the projective spectrum

$$Bl_YX=\operatorname{Proj} (\mathcal{O}_X\oplus I_Y \oplus I_Y^2 \oplus \ldots),$$

which comes with a surjective morphism $\phi:Bl_YX\to X$ (isomorphism everywhere except on $Y$).

The question is the following: the surjective morphism $\phi$ comes as an arrow-reversing map at the level of finitely generated, $\mathbb{Z}$-graded algebras? Since $X$ is projective, I can write $X=\operatorname{Proj}(\bigoplus_{m\geq 0} H^0(X,\mathcal{O}_X(m)))$, but

$$\bigoplus_{m\geq 0} H^0(X,\mathcal{O}_X(m)) \hookrightarrow \bigoplus_{m\geq 0} I_Y^m$$

doesn't look as an inclusion of algebras. I apologize for the probably dumb question but I'm having an hard time understanding the Proj.

$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

The map you want isn't quite what you've written down: for instance, if $X=\Bbb P^2$ and $Y$ is a point, the LHS is $k[x,y,z]$ while the RHS is just $k$. Instead, you want to apply $\Gamma_*(-)=\bigoplus_d \Gamma(-(d))$ to the inclusion of $\mathcal{O}_X$ in to $\bigoplus_m I_Y^m$. The resulting map is injective because $\mathcal{O}_X$ is a direct summand of $\bigoplus_m I_Y^m$.

$\endgroup$
2
  • 1
    $\begingroup$ Dear @KReiser, thank you for answering the question! Before accepting it, I have a few questions, as I don't understand the second part. Why would you want to apply $\Gamma_*$ (can you provide a reference?), and what you mean by $\Gamma(-(d))$ (some sort of shift)? Moreover, why do you want to consider the inclusion of $\mathcal{O}_X$? I warmly thank you again for the patience, have a nice day! $\endgroup$
    – student
    Aug 27, 2021 at 12:03
  • 1
    $\begingroup$ The reason you want to apply $\Gamma_*$ to the inclusion $\mathcal{O}_X\to\bigoplus_m I_Y^m$ is because that gives you an injective map from $\bigoplus_{m\geq 0} H^0(X,\mathcal{O}_X(m))$ to $\bigoplus_{m\geq 0} H^0(X,\bigoplus_d I_Y^d)$, which is the correct version of the statement you want. By $\Gamma(-(d))$ I mean the functor that takes a sheaf $\mathcal{F}$ on $X$ to the global sections of $\mathcal{F}(d)$ - this is the same as taking $H^0(X,\mathcal{F}(d))$, which you're already doing. If you're looking for more material on $\Gamma_*$, check Hartshorne chapter II section 5, for instance. $\endgroup$
    – KReiser
    Aug 27, 2021 at 18:31

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .